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Proceeding of the IEEE International Conference on Automation and Logistics
Zhengzhou, China, August 2012
Stabilization of Singular Fractional-Order Systems·
A Linear Matrix Inequality Approach
Xiaona Songl,2 and Leipo Liu 1 I. Electronic and Information Engineering College
Henan University of Science and Technology
471003 Luoyang, China
2. Luoyang Optoelectro Technology Development Center
471009 Luoyang, China
xiaona_97@ 163.com liuleipo123@yahoo.com.cn
Abstract-In this study, the problems of stability and stabilization for singular fractional-order (SFO) systems have been studied. For the stability problem, conditions are given such that the SFO system is regular and stable; while for the stabilization problem, we design a state feedback control law which guarantees the resulting closed-loop system is stable. In terms of linear matrix inequality, an explicit expression for the desired state feedback control is given. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.
Index Terms-Singular systems, fractional-order systems, linear matrix inequality
I. INTRODUCTION
Singular systems, which are also referred to as descriptor
systems, implicit systems, generalized state-space systems
and so on [7], [9], have been widely studied by many authors
in the past years. This is due not only to the theoretical
interest but also to the extensive application of such system in
large-scale systems, economic systems, power systems, and
other areas [7], [9]. With respect to the problem of stability
and stabilization for the singular systems, much attention
has been focused on the problem of robust stability and
robust stabilization for continuous-time singular systems [5],
[20], [21] and discrete-time singular systems [19], [22]. The
problem of robust stability for interval descriptor systems has
also been presented in [12]. But it should be pointed out that
all the existing results are for normal state space system.
On the other hand, fractional-order (FO) systems have
attracted increasing interest [I], [2], [16], [18]. This is mainly
due to the fact that many real-world physical systems are
better characterized by FO differential equations [15]. The
analysis of stability and stabilization for FO linear time
invariant (FO-LTI) system have been widely investigated, and
there have been many results [8], [II], [13], [17], [23]. For
interval FO-LTI systems, the stability and the controllability
problems have been addressed for the first time in [14] and
[6], respectively. Recently, the stability analysis of FO-LTI
systems with order 1 � a < 2 is converted into the domain
978-1-4673-0364-4112/$31.00 ©2012 IEEE 19
Zhen Wang3
3. College of Information Science and Engineering
Shandong University of Science and Technology
266590 Qingdao, China
wangzhen. sd@gmail.com
of ordinary systems which is well established and well
understood in [18]; while in [16], a necessary and sufficient
LMI condition for stability analysis of FO-LTI system with
order 0 < a < 1 is given. But for singular fractional-order
(SFO) system, the problems of stability and the stabilization
are rarely attacked.
The main contribution of this paper include the analysis of
the stability and stabilization condition for SFO LTI systems.
The Caputo definition for fractional derivative is adopted. The
purpose of the stability problem is to give condition such that
the SFO system is regular and stable; while the aim of the
stabilization is to design a state feedback control law such
that the resulting closed-loop system is regular and stable. In
terms of linear matrix inequality eLMI), sufficient conditions
for the solvability of the stability and stabilization problem
for SFO LTI system of 0 < a < 2 are proposed. When
the LMI is feasible, an explicit expression of a desired state
feedback controller is also given.
Notations: Throughout this paper, for real symmetric
matrices X and Y, the notation X � Y (respectively,
X > Y) means that the matrix X -Y is positive semidefinite
(respectively, positive definite). The notation NIT represents
the transpose of the matrix lVI. Inxn denotes the nxn identity
matrix. In symmetric block matrices, "*" is used as an ellipsis
for terms induced by symmetry. Matrices, if not explicitly
stated, are assumed to have appropriate dimensions. Sym(X) denotes the expression X + XT.
II. PRELIMINARIES AND PROBLEM FORMULATION
In this paper, we adopt the following Caputo definition for
fractional derivative, which allows utilization of initial values
of classical integer-order derivatives with known physical
interpretations [4], [15]
D"f(t) = d"f(t)
=
1 it f(n)(T)dT (I)
dta r(n-a) 0 (t-T)a+l-n'
where n is an integer satisfying n - 1 < a � n.
Considering the following SFO LTI system:
EDCtx(t) = Ax(t) + Bu(t) , 0 < a < 2, (2)
where a is the time fractional derivative order. x(t) ERn is
the state, u(t) E Rrn is the control input. The matrix E E Rnxn is singular, we shall assume that rank E = r < n. A and B are known real constant matrices with appropriate
dimensions.
Without loss of generality, we suppose E, A and B have
the following form
E= [ ci The nominal unforced SFO system of (2) can be written as
EDCtx(t) = Ax(t) . (4)
Definition I : I) The SFO system (4) is said to be regular if det(sCt E-A)
is not identically zero.
2) The SFO system (4) is said to be impulse free if (4) is
regular and deg(det(sCtE - A)) = rankE. Definition II :
The SFO system (2) is said to be stabilizable if there
exists a linear state feedback control law u(t) = Kx(t), K E RTnxn such that the closed-loop system is regular,
impulse free and stable in the sense of Definition I. In this
case,
u(t) = Kx(t), (5)
is said to be a state feedback control law for system (2).
Generally speaking, there are two kinds of stabilization
problems for singular continuous-time systems. One is to
determine state feedback controllers such that the c1osed
loop system is regular, impulse-free and stable. The other is
to design state feedback controllers to make the closed-loop
system regular and stable [22]. In this paper, we deal with
the second stabilization problem for the SFO system (2).
III. MAIN RESULTS
In this section, we give a solution to the stability analysis
and the stabilization problems formulated in the previous
part, by using a strict LMI approach. We first give the
following results which will be used in the proof of our main
results.
Lemma 1: [i8J The FO-LTI system Dqx = Ax (1 � q < 2) is asymptotically stable if and only if the LTI system,
� [ A sin q7r x-
2 - -Acos q;
is asymptotically stable.
Acos q; ] _
Asin q; x, (6)
Lemma 2: [3J Integer order system x(t) = Ax(t) is
asymptotically exponentially stable if and only if there exists
20
a positive definite matrix PES, where S denotes the set of
symmetric matrices, such that:
ATp + PA<O. (7)
Theorem 1: The system (4) with order 1 � a < 2 is
regular and stable if and only if the following conditions are
satisfied ..
I) A4 is invertible.
2) There exits a symmetric matrix P > 0, the following
LMI satisfied
sym{8 Q9 (Ap)} < 0, (8)
where
8= [ sin Ct27r _ cos (�7r (9)
Proof' For the system as follows
EDCtx(t) = Ax(t) . (10)
Let f(s) = 1 sCtE - A I, then we can prove that f(s) is an
analytical function with respect to s on the whole complex
plane. When s is large enough, 1 sCt E - A IY!O 0, it follows
from f (s) is an analytical function that (sCt E - A) -1 exists
almost everywhere on comlex plane [10].
Therefore, det( sO: E-A) is not identically zero, and system
(10) is regular.
Now, we will present the stable condition.
Let
x(t) = [ ����� ] , then, the SFO system (10) can be decomposed as
where
From (12), we can derive
A1X1(t) + A2X2(t) , A3X1(t) + A4X2(t) ,
This, together with (1l), we have
(I l)
(12)
(13)
(14)
It is easy to see that the stability condition of the SFO system
(10) is equivalent to that of the system (14). In view of this,
next we shall find the stability condition of the system (14).
U sing the Lemma 1, the SFO LTI system Dn Xl (t) =
(AI - A2A4l A3)Xl(t) = AX1(t) (1 ::; a < 2) is asymptot
ically stable if and only if the LTI system,
Xl t - - 2 � [ A sin mr
( ) - -Acos (�7f
is asymptotically stable.
A cos 0i27f ] _
A- . n7f Xl(t) , sm2 (15)
So, the stability condition of (10) is finally equivalent to
that of the system (15).
According to Lemma 2, the system (15) is asymptotically
exponentially stable if and only if there exists a positive
definite matrix PES, where S denotes the set of symmetric
matrices, such that:
[ (Ap + pAT) sin (�7f (P AT - Ap) cos (�7f - - 2 < 0
(Ap - pAT) cos n7f ] (AP + PAT) sin (�7f ,
(16)
which is equivalent to there exits P > 0 such that
sym{8 Q9 (Ap)) < 0, (17)
which is the stable condition. This completes the proof.
Now, we are in a position to present a solution to the state
feedback control problem.
Theorem 2: The system (2) with order 1 ::; a < 2 can
be stabilized by the state feedback controller (5), if there
exist the matrices G, L2 and symmetric matrix P > 0, the
following conditions are satisfied
I) A4 + B2K2 is invertible.
2) the following LMI satisfied
where
Proof Let
AlP + B1G - 2L2, L1A3P = L1B2G,
B2G, (A2 + B1K2)(A4 + B2K2) -1.
from (IS), we can derive
where
On the other hand, for the system
EDOi x(t) = Ax(t) + Bu(t) ,
(IS)
(19)
(20)
(21 )
(22)
(23)
(24)
(26)
21
We design the state feedback controller as in Definition II,
and the controller K has the following form:
(27)
Then, using the expression in (3), and (27), the SFO system
(26) can be decomposed as
(AI + B1Kl)Xl(t) +(A2 + B1K2)X2(t) (2S)
o (A3 + B2Kdxl(t) +(A4 + B2K2)X2(t)
From (29), we can derive
This, together with (28), one can obtain
where
Akl Al + B1Kl, Ak2 = A2 + B1K2,
Ak3 A3 + B2Kl, Ak4 = A4 + B2K2,
(29)
(30)
Using the same method in the stability part, according
to the Lemma I, the stability condition of system (31) is
equivalent to that the system as follows:
£1 (t) =
k sm 2 [ A . C>7f -Ak cos (�7f
Ak cos C>27f ] A (t) A . C>7f Xl . ksm2 (32)
Therefore, the stability condition of (26) is finally equivalent
to that of the system (32).
According to Lemma 2, the system (32) is asymptotically
exponentially stable if and only if there exists a positive
definite matrix PES, where S denotes the set of symmetric
matrices, such that:
[ (AkP + PAn sin ';7f (P A[ - AkP) cos (�7f
(AkP - PAn cos ';7f ] (AkP + PAn sin (�7f
which is equivalent to there exits P > ° such that
< 0,
(33)
(34)
which is the same to (24). This completes the proof.
Theorem 3: [l6) Fractional system DVx(t) = Ax(t) of
order ° < v < 1 is rc> asymptotically stable if and only if
there exist positive definite matrices Xl = X{ E cnxn and
X2 = X� E cnxn such that
(35)
where
r = e.i(l-v)-!f. (36)
From Theorem 3, we can obtain the following theorem
easily.
Theorem 4: FO system (4) of order 0 < v < 1 is
roc asymptotically stable if and only if there exist positive
definite matrices Xl = X; E cnxn and X2 = X� E cnxn such that
1) A4 is invertible.
2) The following LMI satisfied
(37)
where
Proof' Using the similar method in the Proof of the
Theorem I and the Theorem 3, we can obtain the condition
directly. This completes the proof.
Theorem 5: SFO system (2) of order 0 < a < 1 can
be stabilized by (5), if there exist the matrices G, K2 and
symmetric matrix X > 0, such that
I) B2 and A4 + B2K2 is invertible
2) The following LMI satisfied
where
G = LA3X, L = (A2+B1K2)(A4 +B2K2)-1, A3 = B2Kl. (40)
Proof' For system
EDOCx(t) = Ax(t) + Bu(t), 0 < a < 1, (41)
using the similar method in the Proof of the Theorem 2, we
can obtain that the system above is asymptotically stable if
and only if the system
D"'Xl(t) = [Akl-Ak2Ak4l Adxl(t) = AkXl(t), 0 < a < 1,
(42)
is asymptotically stable.
According to Theorem 4, and let Xl = X2 = X, we can
obtain that the system (42) is asymptotically stable if and
only if the following matrix inequality satisfied:
1 1f(1-a) [AklX - Ak2Ak4 Ak3X] sin
2 < O. (43)
From (40), we can obtain the following LMI
22
1f(1-a) 1 sin 2
Sym(A1X + B1B:; A3X - 2G) < 0, (44)
which is the stabilization condition. This completes the proof.
IV. NUMERICAL EXAMPLE
Consider the stabilization problem for the SFO system (2)
of order 0 < a < 2 with the following parameters:
a 1.2, Al=
A3 3 � ] , 0
Bl -1 0
0 -1
[ � -1 ] 5 ' A2 = [
A4 = [ -5
-1 �6 ] , ] , [ -2 0
B2 = 0 -1
-5 0 ] , 0 -2
] . Obviously, when u(t) = 0, the system (2) is unstable
because the eigenvalues of A are {-4.9515, -2.2321 + 2.3433i, -2.2321- 2.3433i, 4.4158}, which are outside the
stable area. The purpose is to design a state feedback control
law such that the closed-loop system is stable. Now, using
Matlab LMI Control Toolbox to solve the LMI (18), the
asymptotically stabilizing state-feedback gain matrix is ob
tained as
o -3
0.6873 -2.2901
-0.0023 ] -11.1605 .
While, for the SFO system (2) with the following parameters:
0.8, Al = [ -3 �8
] , A2 = [ -1 -2 ] a 0 -1 -2 '
A3 [ -3 � ] , [ -1 � ] , 0 A4 = -1
Bl [ -1 �1
] , [ -3 0 ] . 0 B2 = 0 -1
when u(t) 0, the system (2) is unstable because the
eigenvalues of A are { 5.1629, 0.7183,-4.4833,-7.3979}. Then, using Matlab LMI Control Toolbox to solve the LMI
(39), we can obtain the state-feedback gain matrix. With the
state feedback controller, the closed-loop system is stable.
The state response of the SFO system of order a = 1.2 are
given in Fig. I, while Fig. 2 shows the state response of the
SFO system of order a = 0.8. From these simulation results, it can be seen the designed
state feedback controller ensures the asymptotic stability of
the SFO system.
V. CONCLUSION
The problems of stability and stabilization for SFO system
have been studied. In terms of LMI, sufficient conditions
for the stability and stabilization of the SFO system have
been established. The proposed state feedback control law
guarantees that the closed-loop system is stable.
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23
-0.5
¥ -1
-1.5 1
-2 _,
-2.5
- 3 oL-------------��------------�,O--------------�,5 Time(s)
Fig.I. State response Xi (t), i= 1,2,3,4. (a= 1.2)
-
---',
1.5
-0.5 oL-------------�--------------�, o--------------�,5 Time(s)
Fig.2. State response Xi(t), i=I,2,3,4. (a=O.8)